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Test-time adaptation (TTA) addresses the machine learning challenge of adapting models to unlabeled test data from shifting distributions in dynamic environments. A key issue in this online setting arises from using unsupervised learning techniques, which introduce explicit gradient noise that degrades model weights. To invest in weight degradation, we propose a Bayesian weight enhancement framework, which generalizes existing weight-based TTA methods that effectively mitigate the issue. Our framework enables robust adaptation to distribution shifts by accounting for diverse weights by modeling weight distributions.Building on our framework, we identify a key limitation in existing methods: their neglect of time-varying covariance reflects the influence of the gradient noise. To address this gap, we propose a novel steady-state adaptation (SSA) algorithm that balances covariance dynamics during adaptation. SSA is derived through the solution of a stochastic differential equation for the TTA process and online inference. The resulting algorithm incorporates a covariance-aware learning rate adjustment mechanism. Through extensive experiments, we demonstrate that SSA consistently improves state-of-the-art methods in various TTA scenarios, datasets, and model architectures, establishing its effectiveness in instability and adaptability.